Math Problems Vault

Problem IV.4.1

Let $\left\{ \mathscr {X}_i : i \in I\right\}$ be a family of LCS's and give $\mathscr {X} = \prod \left\{ \mathscr {X}_i : i \in I\right\}$ the product topology. Show that $L \in \mathscr {X}^*$ if and only if there is a finite subset $F$ contained in $I$ and there are $x_j^*$ in $\mathscr {X}_j^*$ for $j$ in $F$ such that $L(x) = \sum_{j \in F} x_j^*(x(j))$ for each $x$ in $\mathscr {X}$ .

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Problem IV.4.2

Show that the space $s$ of all sequences of scalars (with the metric $d(x,y) = \sum_{n=1}^{\infty } 2^{-n} \left|x(n)-y(n)\right| / (1 + \left|x(n)-y(n)\right|)$ ) is linearly homeomorphic to $C(\mathbb {N})$ and describe $s^*$ .

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Problem IV.4.3

Recall Proposition 4.2: $L \in H(\mathbb {D})^*$ if and only if there is an $r < 1$ and a unique function $g$ analytic on $\mathbb {C}_\infty \setminus \bar{B}(0;r)$ with $g(\infty ) = 0$ such that

L(f) = \frac{1}{2\pi i} \int _\gamma fg

for every $f$ in $H(\mathbb {D})$ , where $\gamma (t) = \rho e^{it}$ , $0 \le t \le 2\pi$ , and $r < \rho < 1$ .

Show that the function $g$ obtained in this proposition is unique.

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Problem IV.4.4

Show that $L \in H(\mathbb {D})^*$ if and only if there are scalars $b_0, b_1, \dots$ in $\mathbb {C}$ such that $\limsup \left|b_n\right|^{1/n} < 1$ and $L(f) = \sum_{n=0}^{\infty } \frac{1}{n!} f^{(n)}(0) b_n$ .

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Problem IV.4.5

If $G$ is an annulus, describe $H(G)^*$ .

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Problem IV.4.6

(a)(b)(c)(d)

(Buck.) Let $X$ be locally compact and let $\beta$ be the strict topology on $C_b(X)$ : for each $\phi$ in $C_0(X)$ , define the seminorm $p_\phi (f) = \left\| \phi f\right\|_\infty$ for $f$ in $C_b(X)$ , and let $\beta$ be the topology defined by these seminorms. Prove the following statements.

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(a)

If $\mu \in M(X)$ and $\varepsilon_n \downarrow 0$ , then there are compact sets $K_1, K_2, \dots$ such that for each $n \ge 1$ , $K_n \subseteq \operatorname {int} K_{n+1}$ and $\left|\mu \right|(X \setminus K_n) < \varepsilon_n$ .

(b)

If $\mu \in M(X)$ , then there is a $\phi$ in $C_0(X)$ such that $\phi \ge 0$ , $\left|\mu \right|(X \setminus \left\{ x : \phi (x) > 0\right\} ) = 0$ , $1/\phi \in L^1(\left|\mu \right|)$ , and $\int 1/\phi \, d\left|\mu \right| \le 1$ .

(c)

Show that if $\mu \in M(X)$ and $L(f) = \int f \, d\mu$ for $f$ in $C_b(X)$ , then $L \in (C_b(X), \beta )^*$ .

(d)

Conversely, if $L \in (C_b(X), \beta )^*$ , then there is a $\mu$ in $M(X)$ such that $L(f) = \int f \, d\mu$ for $f$ in $C_b(X)$ .

Problem IV.4.7

Let $X$ be completely regular and let $\mathscr {M}$ be a linear manifold in $C(X)$ . Show that if for every compact subset $K$ of $X$ , $\mathscr {M} | K \equiv \left\{ f | K : f \in \mathscr {M}\right\}$ is dense in $C(K)$ , then $\mathscr {M}$ is dense in $C(X)$ .

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